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2022　Fundamentals of Probability

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Academic unit or major
Undergraduate major in Mathematical and Computing Science
Instructor(s)
Miyoshi Naoto  Nakano Yumiharu  Yajima Moeko 
Class Format
Lecture / Exercise    (Blended)
Media-enhanced courses
Day/Period(Room No.)
Mon7-8(W935)  Thr5-8(W935)  
Group
-
Course number
MCS.T212
Credits
3
Academic year
2022
Offered quarter
2Q
Syllabus updated
2022/4/4
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

This course emphasizes that students learn the basic skills of probabilistic representation of random phenomena and gives lectures on fundamental concepts of probability theory. The course also facilitates students' understanding by giving exercises and assignments.

Student learning outcomes

Students will be able to acquire the basic skills of mathematical representation for probabilistic phenomena.

Keywords

Probability space, Independence and conditional probability, Random variables and their distributions, Lows of large numbers, Central limit theorem

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

Class flow

Two 100 minute lectures and one 100 minute exercise per week.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction to Probability Understand the necessity for the idea of probability.
Class 2 Probability Spaces and Fundamental Properties of Probability Understand the definition of probability spaces and their fundamental properties.
Class 3 Exercises regarding the contents covered up to the 2nd lecture Cultivate more practical understanding by doing exercises.
Class 4 Absolutely Continuous Distributions and Distribution Functions Understand probability distribution functions and the notion of absolute continuity.
Class 5 Random Variables and Measurable Functions Understand the definitions of random variables and measurable functions.
Class 6 Exercises regarding the contents covered up to the 5th lecture Cultivate more practical understanding by doing exercises.
Class 7 Probability Distributions of Random Variables Understand the probability distributions of random variables.
Class 8 Convergences of Random Variables and Distributions Understand the notions of convergences of random variables and their distributions.
Class 9 Exercises regarding the contents covered up to the 8th lecture Cultivate more practical understanding by doing exercises.
Class 10 Expectations Understand the definition of expectations.
Class 11 Distributions of Random Variables and Expectations Understand the relation of the distributions of random variables and the expectations.
Class 12 Exercises regarding the contents covered up to the 11th lecture Cultivate more practical understanding by doing exercises.
Class 13 Variances, Covariances and Moments Understand the definitions of variances, covariances, and moments.
Class 14 Convergence Theorems for Expectations Understand some convergence theorems for expectations.
Class 15 Exercises regarding the contents covered up to the 14th lecture Cultivate more practical understanding by doing exercises.
Class 16 Probability Generating Functions and Moment Generating Functions Understand the definition of probability generating functions and moment generating functions.
Class 17 Characteristic Functions Understand the definitions of characteristic functions.
Class 18 Exercises regarding the contents covered up to the 17th lecture Cultivate more practical understanding by doing exercises.
Class 19 Law of Large Numbers Understand the law of large numbers.
Class 20 Central Limit Theorem Understand the central limit theorem.
Class 21 Exercises regarding the contents covered up to the 20th lecture Cultivate more practical understanding by doing exercises.
Class 22 Final Exam. Check the level of understanding through the final exam.

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to reference books and other course material.

Textbook(s)

Not required.

Reference books, course materials, etc.

Nishio, Makiko. Probability Theory. Jikkyo-Shuppan. (Japanese)
Ito, Kiyoshi. Fundamentals of Probability Theory. Iwakura-Shoten. (Japanese)
Shiga, Tokuzo. From Lebesgue Integrals to Probability Theory. Kyoritsu-Shuppan. (Japanese)
Kumagaya, Takashi. Probability Theory. Kyoritsu-Shuppan. (Japanese)
Takahashi, Yukio. Probability Theory. Asakura-Shoten. (Japanese)

Assessment criteria and methods

Scores are based on final exam, exercise problems and assignments.

Related courses

  • MCS.T312 ： Markov Analysis
  • MCS.T333 ： Information Theory
  • MCS.T223 ： Mathematical Statistics
  • XCO.B103 ： Foundations of Computing 3
  • MCS.T332 ： Data Analysis
  • MCS.T304 ： Lebesgue Interation

Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites, but it is preferable to study Foundations of Computing 3 (XCO.B103).

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